2026 Summer Experiences in Mathematical Sciences
The Department of Mathematical Sciences is happy to offer an opportunity for research and study in mathematics in the summer of 2026. This program (SEMS) is offered free of charge to Carnegie Mellon undergraduate students who satisfy the following eligibility requirements:
- Students have to be Carnegie Mellon undergraduate students majoring in mathematics.
- Students have to be in good standing.
- Students who participated in SEMS in previous years are not eligible to apply again.
- Students who have already made serious time commitments in the form of participating in internships, taking more than 12 units of summer courses or working as teaching assistants in the summer should not apply for this program.
- Students who participate in SEMS 2026 are expected to return to campus in the fall and give a small presentation of their work at the SEMS symposium that will take place in October.
- Students who are receiving funding for summer research (through the SURF program or through faculty research grants) are not eligible to apply for SEMS.
Interested students must apply in order to participate. Acceptance into any particular SEMS project is not guaranteed. Furthermore, the department may cancel projects that do not generate sufficient demand. Students can join at most one research group. Once accepted, students are expected to devote at least 20 hours a week to any research project they join. Note that SEMS does not provide funding for students.
Participating in SEMS qualifies students for SURA credit. Note that the SURA credit is not automatic. To receive this credit, students will need to enroll in SURA. See this page for instructions on how to enroll in SURA.
Applications for SEMS 2026 are now closed.
Students who are accepted will be notified by April 25. Students who participate in SEMS are expected to attend the Undergraduate Summer Research Seminar in-person or on Zoom. The Undergraduate Summer Research Seminar will have one meeting each week from June 4 until July 2.
➤ Problems in Ramsey Theory
Advisors: Thomas Bohman and Quentin Dubroff
Ramsey theory is the study of ordered structures within large mathematical systems. For example, in classical graph Ramsey theory we are looking for the largest monochromatic complete graph in an arbitrary 2-coloring of the edges of a large complete graph. We will study variations of this question in the context of colorings of the elements of a partially ordered set or the edges of a hypergraph.
Prerequisites: 21-228 Discrete Mathematics, or 21-301 Combinatorics, or 21-484 Graph Theory, or 21-701 Discrete Mathematics.
June 1 - July 24
Modality: In-person.➤ How do deterministic jumps affect the mixing of random walks?
Advisor: Gautam Iyer
Consider a random walk which takes a deterministic jump, and then chooses a nearest neighbor at random. How does the deterministic jump affect the rate of convergence? Recent papers have shown that a "large fraction" of deterministic jump paths will increase the rate of convergence by an order of magnitude (and "speed up mixing"). However, it is also possible that certain jump paths slow down mixing. Our aim is to study such random walks and better understand the effect of the deterministic jump on the mixing rate.
Prerequisites: 21-326 Markov Chains: Theory, Simulation and Applications, or equivalent, with grade A.
June 22 - August 7
Modality: In-person.➤ Computational aspects of low-dimensional convex geometry
Advisor: Konstantin Tikhomirov
Two- and three-dimensional convex geometry deals with problems concerning convex regions in the plane and convex sets in $R^3$, focusing on properties of arrangements of convex sets and their interactions. These include various notions of packings and coverings, as well as properties of contact and intersection graphs generated by families of convex sets.
The increased computational power and the reduced cost of code development due to advances in LLMs make certain long-standing questions amenable to resolution, or at least substantial progress, through computational approaches. The goal of the proposed project is to make progress on several such problems through an efficient combination of mathematical (proof-based) reasoning and computation. Specific topics may include approximation of convex bodies in $R^3$ by polytopes, lattice packings and coverings in $R^3$, and illumination problems for certain classes of convex sets.
Prerequisites: (a) Solid programming skills (the actual coding will likely be done in C or C++, however, solid programming experience using other languages may also be sufficient); (b) ability to read and produce mathematical proofs; (c) solid understanding of two- and three-dimensional Euclidean geometry and linear algebra. The applicants are expected to be familiar with the basic concepts of object-oriented programming and have experience working with multidimensional arrays and tree data structures. The applicants should be prepared to take a short math/programming test during the interview.
May 11 - June 26
Modality: In-person, except for one week of zoom discussions due to advisor's traveling.